Spanning Trees and Their Modulus

Heather Heier February 1, 2016

Abstract This paper is focused on the concept of modulus applied to rooted spanning trees of a simple, undi- rected graph. It introduces modulus and its importance to analyzing graphs. A simple algorithm is established to quickly calculate the modulus of a tree. This leads to the theory that the modulus of a rooted spanning tree is always less than or equal to the degree of the root. Lastly, modulus is applied to Hamiltonian paths, which initiates the theory of a relationship between the presence of a Hamiltonian path and a minimum modulus spanning tree.

1 Introduction 1.1 Modulus and modulus of trees Modulus is, in general, a way to classify and analyze the connectivity of nodes and the various walks, or paths, from node to node in a graph.[1] Modulus can be applied to acyclic connected graphs which are known as ‘trees’. Although finding the modulus can be computationally expensive, some modulus computations have efficient algorithms. One particular example is the modulus of the family of walks beginning at the root of a tree and terminating at a leaf. To apply modulus, we choose a starting and ending node, and then calculate the modulus to determine how necessary the nodes and edges are to the connectivity of the graph.

Algorithm 1 Compute Modulus of a Tree 1: for each leaf l do 2: set Mod(l) = +∞ 3: end for 4: for all other nodes v do 5: set Mod(v) = ‘unknown’ 6: end for 7: while at least one node has Mod = ‘unknown’ do 8: choose a node v such that Mod(v) = ‘unknown’ and Mod(c) = ‘known’ for all children c of v P Mod(c) 9: set Mod(v) = 1+Mod(c) c a child of v 10: end while ∞ 11: Remark: For convenience, we define 1+∞ = 1

Example 1. Computing Modulus for a Tree.

This is an application of Algorithm 1 to a rooted tree containing two children, c, of the root, r, in which c is the predecessor to four terminal children. The resulting modulus is 4/3 as shown in the top node, known as the root.

1 4/3

2/3 2/3

2 2

1 1 1 1

∞ ∞ ∞ ∞

Figure 1: Computing Modulus

1.2 Spanning trees and their modulus Spanning trees are sub-graphs of a main network that is simply connected and acyclic. These trees are each unique and consist of just enough edges to connect every node once. The collection of spanning trees and their modulus can be used to evaluate the main network. The difficulty with computing the modulus for all spanning trees is due to the possibility of accumulating high quantities of spanning trees as a result of the size of the original graph. In other words, the more complex the original graph in question, the larger the output of possible spanning trees. This raised the question, “Is there a way to find the minimum and maximum modulus spanning trees without computing all of the spanning trees?”

2 Results

Example 1. Spanning Tree Modulus Graph Evaluation

Figure 2: Simple 5 Node and 7 Edge Rooted Graph

Figure 2 illustrates a simple rooted graph with the root indicated by the black node. To begin analyzing the connectivity of this graph, first discover all of the 21 possible spanning trees. See Figure 3. Next, apply Algorithm 1 to each spanning tree to calculate and determine the maximum and minimum modulus paths in the graph. This simple graph required calculations for 21 spanning trees. What if the graph is much larger? There- fore, what are some other ways to evaluate Figure 2? Definition 1. The degree of a node is the number of edges connecting the node to another in the graph.

2 The degree of a node is a classification of the connectivity of that node to the graph. The larger the amount of edges reaching the selected node, the higher the degree. For instance, in Figure 2, the root has a degree of 2 because it is connected to the graph by two edges.

Theorem 1. Let Tr be a tree rooted at r. Then

Mod(Tr) ≤ deg(r).

Proof. For rooted trees, there are two cases. Let n be the total number of children of r. In case one, Tr contains only terminal children, c. By Algorithm 1, Mod(Tr,c) = 1 for each child, making Mod(Tr) = n. The number of edges connecting r to its children is deg(r) by Definition 1. Since each child is connected to the root once, deg(r) = n. Therefore proving Mod(Tr) = deg(r). In case two, let c be a child of r. Temporarily disconnect c from r and let Tc be the resulting tree rooted at c. By using Algorithm 1, we can calculate Mod(Tc). The maximum value obtainable with this algorithm Mod(Tc) is Mod(Tc) = 1. Since < 1 for each such child c, it follows that Mod(Tr) is no larger than the 1+Mod(Tc) number of children of r, thus Mod(Tr) ≤ deg(r). Definition 2. A Hamiltonian path is a path in a simple graph containing every node, while a Hamiltonian cycle is a cycle that traverses every edge once. [2] Theorem 2. Let G = (V,E) be a simple graph, and let r ∈ V be a node. If there exists a Hamiltonian path for G starting at r, then a spanning tree, Tr, rooted at r, is a minimum modulus spanning tree if and only if Tr is a Hamiltonian path starting from r. Proof. Suppose a Hamiltonian path starting at r exists. Then, every Hamiltonian path starting at r has modulus 1/(n − 1). Hence, the theorem follows once we show that a path of n nodes has strictly smaller modulus than any other rooted tree on n nodes. This can be seen as follows. Suppose Tr is a n-node tree rooted at r and that Tr has more than one leaf (i.e. Tr is not a path). Let x and y be two distinct leaves in Tr, and let z be their lowest common ancestor. 0 Let w be the child of z that is an ancestor of x (or is x itself), and let Tr be the tree defined by removing 00 the subtree Tw rooted at w from Tr. Let Tr be the tree obtained by reattaching Tw so that w is the child 0 00 of y. We claim that Mod(Tr) > Mod(Tr) > Mod(Tr ). Since z has at least two children, the first inequality follows because removing the Tw strictly decreases the modulus of the subtree rooted at z. This decrease propagates up the tree (as seen from Algorithm 1) 0 implying that Mod(Tr) > Mod(Tr). The second inequality follows from the fact that, as a leaf, y is given modulus +∞ by the algorithm in 0 00 Tr, but has finite modulus in Tr , since Tw has been attached beneath it. Again, this decrease in modulus propagates up to the root r.

References

[1] Albin, N., Poggi-Corradini, P., Sahneh, F. D., and Goering, M. Modulus of families of walks on graphs. arXiv preprint arXiv:1401.7640 (2014). [2] Diestel, R. Graph Theory, vol. 173 of Graduate Texts In Mathematics. Springer, New York, New York, 1997.

3 Mod = 5/3 Mod = 4/3 Mod = 4/3 Mod = 4/3

Mod = 1 Mod = 3/4 Mod = 3/5 Mod = 3/5

Mod = 3/5 Mod = 3/5 Mod = 3/5 Mod = 3/5

Mod = 2/5 Mod = 2/5 Mod = 2/5 Mod = 1/4

Mod = 1/4 Mod = 1/4 Mod = 1/4 Mod = 1/4

Mod = 1/4

Figure 3: Simple 5 Node and 7 Edge Graph Spanning Trees

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